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Linear combinations of two GPs

Authors
  • avatar
    Name
    Vu Hung
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Problem Statement

Let Tn=arn−1T_n=ar^{n-1} and Un=ARn−1U_n=AR^{n-1}. Define Wn=Tn+UnW_n=T_n+U_n.

  • Show (Wn)(W_n) satisfies
Wn+2−(r+R)Wn+1+rRWn=0.W_{n+2}-(r+R)W_{n+1}+rRW_n=0.
  • Solve
Xn+2−5Xn+1+6Xn=0,X1=5, X2=13.X_{n+2}-5X_{n+1}+6X_n=0,\quad X_1=5,\ X_2=13.

Hints

For (ii), the characteristic roots are 22 and 33.

Solutions

Substitute Wn=arn−1+ARn−1W_n=ar^{n-1}+AR^{n-1} directly to verify (i). For (ii), write

Xn=α2n−1+β3n−1.X_n=\alpha 2^{n-1}+\beta 3^{n-1}.

Using X1=5, X2=13X_1=5,\ X_2=13:

α+β=5,2α+3β=13\alpha+\beta=5,\quad 2\alpha+3\beta=13

gives α=2, β=3\alpha=2,\ \beta=3. Hence

Xn=2n+3n.X_n=2^n+3^n.

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